The circle with centre `(3/2,1/2)` and radius `sqrt(3/2)` is

Options

A. `x^(2)+y^(2)-2x-2y+1=0`
B. `x^(2)+y^(2)+2x+2y+1=0`
C. `x^(2)+y^(2)-3x-y+1=0`
D. `x^(2)+y^(2)+3x+y-1=0`

Answer

To find the equation of the circle with center \((\frac{3}{2}, \frac{1}{2})\) and radius \(\sqrt{\frac{3}{2}}\), we will use the center-radius form of the equation of a circle, which is given by:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

where \((h, k)\) is the center of the circle and \(r\) is the radius.

### Step-by-step Solution:

1. **Identify the center and radius**:
- The center \((h, k)\) is given as \((\frac{3}{2}, \frac{1}{2})\).
- The radius \(r\) is given as \(\sqrt{\frac{3}{2}}\).

2. **Substitute the values into the center-radius form**:
- Substitute \(h = \frac{3}{2}\), \(k = \frac{1}{2}\), and \(r = \sqrt{\frac{3}{2}}\) into the equation:
\[ (x - \frac{3}{2})^2 + (y - \frac{1}{2})^2 = (\sqrt{\frac{3}{2}})^2 \]

3. **Calculate \(r^2\)**:
- Calculate \(r^2\):
\[ r^2 = \left(\sqrt{\frac{3}{2}}\right)^2 = \frac{3}{2} \]
- Thus, the equation becomes:
\[ (x - \frac{3}{2})^2 + (y - \frac{1}{2})^2 = \frac{3}{2} \]

4. **Expand the left-hand side**:
- Expand \((x - \frac{3}{2})^2\):
\[ (x - \frac{3}{2})^2 = x^2 - 2 \cdot x \cdot \frac{3}{2} + \left(\frac{3}{2}\right)^2 = x^2 - 3x + \frac{9}{4} \]
- Expand \((y - \frac{1}{2})^2\):
\[ (y - \frac{1}{2})^2 = y^2 - 2 \cdot y \cdot \frac{1}{2} + \left(\frac{1}{2}\right)^2 = y^2 - y + \frac{1}{4} \]

5. **Combine the expansions**:
- Combine both expansions:
\[ x^2 - 3x + \frac{9}{4} + y^2 - y + \frac{1}{4} = \frac{3}{2} \]

6. **Combine constant terms**:
- Combine the constant terms on the left:
\[ \frac{9}{4} + \frac{1}{4} = \frac{10}{4} = \frac{5}{2} \]
- Thus, the equation becomes:
\[ x^2 + y^2 - 3x - y + \frac{5}{2} = \frac{3}{2} \]

7. **Rearranging the equation**:
- Move \(\frac{3}{2}\) to the left side:
\[ x^2 + y^2 - 3x - y + \frac{5}{2} - \frac{3}{2} = 0 \]
- Simplifying gives:
\[ x^2 + y^2 - 3x - y + 1 = 0 \]

### Final Equation:
The equation of the circle is:
\[ x^2 + y^2 - 3x - y + 1 = 0 \]